Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

20 1. Newtonian turbulence
Figure 1.3: Probability distributions, normalized with their standard deviations
σr =< δu 2 r >1/2 , of transverse velocity increments in a turbulent air jet at Rλ =
695 for different separations: r ∼ η (dotted line); r ∼ ℓ0 shifted of three decades
(long dashed line); intermediate scales (solid line) [13].
A phenomenological model of intermittency is the multifractal one, introduced
by Parisi and Frisch [14]. In the multifractal approach [14, 15], instead of global
scale invariance, local scale invariance is assumed; more specifically, hypothesis
H2 is modified into:
HMF in the limit Re → ∞, the turbulent flow possesses a range of scaling
exponents I = [hmin, hmax]. For all h ∈ I, there is a set Sh ⊂ R3 of
dimension D(h), such that for x ∈ I and ℓ → 0:
� �h δuℓ(x) ℓ
∼
(1.60)
u0
where, of course, u0 is the large scale velocity.
The structure function of order p is then expressed as a superposition, weighted
with a measure dµ(h), of different contributions originated by different scaling
exponents:
Sp(ℓ)
u p
0
�
∼
I
ℓ0
� �ph+3−D(h) ℓ
dµ(h)
ℓ0
(1.61)
where the factor (ℓ/ℓ0) 3−D(h) is the probability of being within a distance ℓ of the
set Sh. The exponent ζp can be obtained in the limit ℓ → 0 by a saddle-point
20